Strictly single-site DMRG algorithm with subspace expansion

We introduce a strictly single-site DMRG algorithm based on the subspace expansion of the alternating minimal energy (AMEn) method. The proposed new MPS basis enrichment method is sufficient to avoid local minima during the optimization, similar to the density matrix perturbation method, but computationally cheaper. Each application of $\widehat{H}$ to $|\Psi \rangle$ in the central eigensolver is reduced in cost for a speed-up of $\approx (d+1)/2$, with $d$ the physical site dimension. Further speed-ups result from cheaper auxiliary calculations and an often greatly improved convergence behavior. Runtime to convergence improves by up to a factor of 2.5 on the Fermi-Hubbard model compared to the previous single-site method and by up to a factor of 3.9 compared to two-site DMRG. The method is compatible with real-space parallelization and non-Abelian symmetries.

Figure 1

Energies of the state at different points during a single update: before optimization, the state has some initial energy $E_i$. Local optimization via the eigensolver takes this energy down by $\Delta E_O$ to $E_{\min }$. Subsequent truncation causes a rise in energy by $\Delta E_T$ with the final value at the end of this update being $E_f$.

Figure 2

Quantum number distribution as counted from the right at each bond of a $l=20$ system with $S=\frac{1}{2}$ and $S_z^{\mathrm{total}}=0$. The artificial input state is shown with black circles. Two DMRG calculations have then been done on this input state, once with no enrichment term ($\alpha =0$, red squares) and once with subspace expansion enabled ($\alpha \ne 0$, blue diamonds). It is clearly visible that without enrichment, DMRG3S can reduce some weights to zero, but cannot add new states—red only occurs together with black. As soon as enrichment is enabled, DMRG3S restores $\pm S_z$ symmetry and reflective symmetry over the $10\mathrm{th}$ bond and finds a much better ground state.

Figure 3

Spin chain Eq. (40): normalized error in energy as a function of sweeps (left) and CPU time used (right) of the two single-site algorithms at different $m=200,400,800$. DMRG3S shows both a speed-up and an improved convergence per sweep compared to CWF, with a long tail of slow convergence very visible for CWF at high accuracies.

Figure 4

Bosonic system Eq. (41): normalized error in energy from CWF and DMRG3S as a function of sweeps (left) and CPU time used (right) for $m=50,100,200$. Again, an improved convergence behavior at high accuracies can be observed, in particular at smaller values of $m$. The small bond dimensions lead to a smaller speed-up due to faster numerical operations, which only becomes visible at $m=200$.

Figure 5

Fermi-Hubbard Eq. (42): normalized error in energy from DMRG3S and CWF as a function of sweeps (left) and CPU time used (right) for different bond dimensions $m=300,600,1200$. The same basic behavior as for the previous systems is repeated, with both improved convergence behavior at high accuracies and faster numerical operations.

Figure 6

Free fermions Eq. (43): normalized error in energy from CWF and DMRG3S as a function of sweeps (left) and CPU time used (right) at $m=300,600,1200$. CWF again exhibits a long tail of slow convergence, while DMRG3S converges quickly at all $m$ and all accuracies.